On matrix variance inequalities

Afendras, Georgios; Papadatos, Nickos

doi:10.1016/j.jspi.2011.05.016

Cited by 7 publications

(27 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The conditions appearing in the definition of F(A ,η p ) are tailored to ensure that all identities and manipulations follow immediately. For instance, the requirement that L p η(·)g(· − ) ∈ F (1) (p) in the definition of F(A ,η p ) guarantees that the resulting functions A ,η p g(x) have p-mean 0 and the condition (L p η)∆ − g ∈ L 1 (p) guarantees that the expectations of the individual summands on the r.h.s. of (2.7) exist.…”

Section: Stein Operators and Stein Equationsmentioning

confidence: 99%

First-order covariance inequalities via Stein’s method

Ernst¹,

Reinert²,

Swan³

2020

Bernoulli

View full text Add to dashboard Cite

We propose probabilistic representations for inverse Stein operators (i.e. solutions to Stein equations) under general conditions; in particular we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and non-uniform Stein factors (i.e. bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases, and expressed in terms of objects familiar within the context of Stein's method. Applications of the Cauchy-Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed. * Université de Liège, corresponding author Yvik Swan: yswan@uliege.be.

show abstract

Section: Stein Operators and Stein Equationsmentioning

confidence: 99%

First-order covariance inequalities via Stein’s method

Ernst¹,

Reinert²,

Swan³

2020

Bernoulli

View full text Add to dashboard Cite

show abstract

“…Chernoff proved that

V a r 1 (Z) \leq E {(1' (Z))}^{2}

, provided that

E {(1' (Z))}^{2}

is finite, where the equality holds iff g is a linear function; see also the previous papers by Nash , Brascamp and Lieb . This inequality has been generalized and extended by many authors (see, e.g., ).…”

Section: Introductionmentioning

confidence: 89%

A note on a variance bound for the multinomial and the negative multinomial distribution

Afendras

Papathanasiou

2014

Naval Research Logistics

View full text Add to dashboard Cite

show abstract

“…As mentioned in the introduction, a matrix extension of Chernoff's gaussian bound (1.16) is due to [59], and the result is obtained by expanding the test functions in the Hermite basis. An extension of this result to a wide class of multivariate densities is proposed in [2] (see also references therein). Once again, our notations allow to extend this result to arbitrary densities of real-valued random variables.…”

Section: Olkin-shepp-type Boundsmentioning

confidence: 96%

“…More recently, the contributions [4], [5] [3] and [1] and [51,52] begin to fully explore connections with Stein's method. In the Gaussian framework, an enlightening first order matrix variance bound is proposed in [59] for the Gaussian distribution, and in a more general setting in [2] (and also to arbitrary order). Finally, we mention [66,65] and [64]'s revisiting of this classical literature enticed us to begin the work that ultimately led to the present paper.…”

Section: Some Referencesmentioning

confidence: 99%

Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions

Ernst,

Reinert,

Swan

2018

Preprint

View full text Add to dashboard Cite

In this paper, following on from [49,50,63] we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use for revisiting the very classical topic of bounding the variance of functionals of random variables. Applying the Cauchy-Schwarz inequality yields first order upper and lower Klaassen [45]-type variance bounds. A probabilistic representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou [60]-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp [59] type covariance bounds. All results hold for univariate target distribution under very weak assumptions (in particular they hold for continuous and discrete distributions alike). Many concrete illustrations are provided.

show abstract

On matrix variance inequalities

Cited by 7 publications

References 7 publications

First-order covariance inequalities via Stein’s method

First-order covariance inequalities via Stein’s method

A note on a variance bound for the multinomial and the negative multinomial distribution

Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions

Contact Info

Product

Resources

About